Remark 4.1. The representations (REF ) mean that this
is a version of the Galerkin method. However, unlike classical well-posed
forward problems for PDEs, where Galerkin method is used and its convergence
at \(N\rightarrow \infty \)  is usually proven, we cannot prove
convergence of our inversion numerical procedure described below for \(N\rightarrow \infty \). This is basically because of the ill-posed
nature of our CIP. Thus, we actually work below within the framework of an
approximate mathematical model. Then, however, the question can be raised
whether this model really works numerically. The answer is positive, and
this answer is obtained computationally in section 7. We observe that very
similar truncated series were used in some other above cited works on the
convexification, such as, e.g. [1]}, [2]}, [3]}, [4]}, and all of them have demonstrated good numerical performances,
including the quite challenging cases of experimentally collected data for
backscatter microwaves [2]}, [4]} . Likewise,
truncated Fourier series were used in works of other authors about CIPs,
such as, e.g. [7]}, [8]}, [9]}, [10]} and also without proofs of convergence
of inversion procedures at \(N\rightarrow \infty .\) Those proofs were
not provided for the same reason as the one here: the ill-posed nature of
CIPs. Finally, we refer to subsection 3.4 of [3]} for a detailed
discussion of this issue for the case of a similar CIP for the conventional
RTE.
